Emergent Behaviors of Bonabeau Swarms

"Intuition can be a particularly poor guide to prediction of the behavior of complex systems above a few levels of complexity" [1]. Bonabeau and Myers [2] initially posed the Bonabeau swarm problems to illustrate how simple rules, when applied by many individual agents simultaneously, can result in a complex and sometimes unpredictable emergent behavior. Gravagne & Marks [3] describe the swarm behaviors that can occur in Bonabeau swarms by assigning a particular dynamical interpretation to the swarm rules and examining the properties of the corresponding dynamics. To begin we first note examples of the simple rules that each swarm agent will enact.

A. The Peacemaker: Every agent in the swarm picks two others and tries to position itself between them.

B. The Coward: Every agent in the swarm picks two others and tries to position itself so that the first is directly between itself and the second.

C. The Follower: Every agent in the swarm picks another and chases it.

Each of these three examples can be construed as special cases of these dynamic equations.
The emergent behaviors of these swarms, as empirically observed by Bonabeau and Myers [2] are analytically verified by Gravagne & Marks [3]

The Peacemaker: Cluster

There are 1000 agents on a 500x500 field. The step size is gamma = 0.01 and alpha=0.5.

For alpha=1, each agent chooses another and follows. There are 1000 agents on a 500x500 field.
The snapshot example is disconnected into three swarms. This is evident in that convergence is to three points.
The avi movie also has disconnedted swarms.
The step size is gamma = 0.01 for the snapshots and 0.02 for the movie.